Question: The lifespans of turtles in a particular zoo are normally distributed. The average turtle lives $91$ years; the standard deviation is $21.4$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a turtle living between $69.6$ and $155.2$ years.
Answer: $91$ $69.6$ $112.4$ $48.2$ $133.8$ $26.8$ $155.2$ $99.7\%$ $68\%$ $15.85\%$ $15.85\%$ We know the lifespans are normally distributed with an average lifespan of $91$ years. We know the standard deviation is $21.4$ years, so one standard deviation below the mean is $69.6$ years and one standard deviation above the mean is $112.4$ years. Two standard deviations below the mean is $48.2$ years and two standard deviations above the mean is $133.8$ years. Three standard deviations below the mean is $26.8$ years and three standard deviations above the mean is $155.2$ years. We are interested in the probability of a turtle living between $69.6$ and $155.2$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the turtles will have lifespans within 3 standard deviations of the average lifespan. It also tells us that $68\%$ of the turtles will have lifespans within 1 standard deviation of the mean. The probability of a particular turtle living between $69.6$ and $155.2$ years is ${68\%} + \color{orange}{15.85\%}$, or $83.85\%$.